Meantone family
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[[toc]] The [[5-limit]] parent [[Comma|comma]] of the [[meantone]] family is the Didymus or [[http://en.wikipedia.org/wiki/Syntonic_comma|syntonic comma]], 81/80. This is the one they all temper out. The [[Monzos and Interval Space|monzo]] for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval. [[POTE tuning|POTE generator]]: 696.239 [[Map]]: [<1 0 -4|, <0 1 4|] EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]] [[Badness]]: 0.00736 ==Seven limit children== The [[7-limit]] children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1>, |-13 10 0 -1>], flattone, with normal list [|-4 4 -1>, |-17 9 0 1>], dominant, with normal list [|-4 4 -1>, |6 -2 0 -1>], sharptone, with normal list [|-4 4 -1>, |2 -3 0 1>], injera, with normal list [|-4 4 -1>, |-7 8 0 -2>], mohajira, with normal list [|-4 4 -1>, |-23 11 0 2>], godzilla, with normal list [|-4 4 -1>, |-4 -1 0 2>], mothra, with normal list [|-4 4 -1>, |-10 1 0 3>], squares, with normal list [|-4 4 -1>, |-3 9 0 -4>], and liese, with normal list [|-4 4 -1>, |-9 11 0 -3>]. =Septimal meantone= The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the [[7_4|7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and [[7_5|7/5]], C-F#, the tritone. The [[Wedgies and Multivals|wedgie]] for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and [[31edo]] is a good tuning for it. [[Comma]]s: 81/80, 126/125 7 and [[9-limit]] minimax [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 696.495 Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly. [[Map]]: [<1 0 -4 -13|, <0 1 4 10|] [[Generator]]s: 2, 3 [[Wedgie]]: <<1 4 10 4 13 12|| EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]] [[Badness]]: 0.0137 ==Unidecimal meantone aka Huygens== [[Comma]]s: 81/80, 126/125, 99/98 [[11-limit]] minimax [|1 0 0 0 0>, |25/16 -1/8 0 0 1/16>, |9/4 -1/2 0 0 1/4>, |21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>] [[Eigenmonzo]]s: 2, 11/9 [[POTE tuning|POTE generator]]: 696.967 [[Algebraic generator]]: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents. [[Map]]: [<1 0 -4 -13 -25|, <0 1 4 10 18|] [[Generator]]s: 2, 3 EDOs: [[7edo|7]], [[12edo|12]], [[31edo|31]], [[105edo|105]], [[198edo|198]] [[Badness]]: 0.0170 ===Tridecimal meantone=== [[Comma]]s: 66/65, 81/80, 99/98, 105/104 [[POTE tuning|POTE generator]]: ~3/2 = 696.642 Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[267edo|267]], [[298edo|298]] [[Badness]]: 0.0180 ===Grosstone=== Commas: 81/80, 99/98, 126/125, 144/143 POTE generator: ~3/2 = 697.264 Map: [<1 0 -4 -13 -25 29|, <0 1 4 10 18 -16|] EDOs: 12, 31, 43, 74 Badness: 0.0259 ==Meanpop== [[Comma]]s: 81/80, 126/125, 385/384 [[11-limit]] [[minimax]] 1/4 comma [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |-3 0 5/2 0 0>, |11 0 -13/4 0 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 696.434 [[Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge. Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] [[Generator]]s: 2, 3 EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]], [[112edo|112]] [[Badness]]: 0.0215 ===13-limit Meanpop=== [[Comma]]s: 81/80, 105/104, 144/143, 196/195 POTE generator: ~3/2 = 696.211 Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] EDOS: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[131edo|131]] [[Badness]]: 0.0209 ==Meanenneadecal== [[Comma]]s: 45/44, 56/55, 81/80 [[POTE tuning|POTE generator]]: ~3/2 = 696.250 Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|] EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]] [[Badness]]: 0.0214 ===13-limit=== [[Comma]]s: 45/44, 56/55, 78/77, 81/80 [[POTE tuning|POTE generator]]: ~3/2 = 696.146 Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|] EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[131edo|131]], [[181edo|181]] [[Badness]]: 0.0212 =Flattone= [[Comma]]s: 81/80, 525/512 The [[wedgie]] for flattone is <<1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that [[7_4|7/4]] is a diminished minor seventh interval. Other intervals are [[7_6|7/6]], a diminished minor third, and [[7_5|7/5]], a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]]. [[7-limit]] minimax [|1 0 0 0>, |21/13 0 1/13 -1/13>, |32/13 0 4/13 -4/13>, |32/13 0 -9/13 9/13>] [[Eigenmonzo]]s: 2, 7/5 [[9-limit]] minimax [|1 0 0 0>, |17/11 2/11 0 -1/11>, |24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>] [[Eigenmonzo]]s: 2, 9/7 [[POTE tuning|POTE generator]]: 693.779 Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4. Map: [<1 0 -4 17|, <0 1 4 -9|] [[Wedgie]]: <<1 4 -9 4 -17 -32|| [[Generator]]s: 2, 3 EDOs: [[7edo|7]], [[19edo|19]], [[45edo|45]], [[64edo|64]] [[Badness]]: 0.0386 =Dominant= [[Comma]]s: 36/35, 64/63 The wedgie for dominant is <<1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3_2|3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]]. [[POTE tuning|POTE generator]]: 701.573 Map: [<1 0 -4 6|, <0 1 4 -2|] [[Wedgie]]: <<1 4 -2 4 -6 -16|| EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[53edo|53]], [[65edo|65]] [[Badness]]: 0.0207 =Sharptone= [[Comma]]s: 21/20, 28/27 Sharptone, with a wedgie <<1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done. [[POTE tuning|POTE generator]]: 700.140 Map: [<1 0 -4 -2|, <0 1 4 3|] [[Wedgie]]: <<1 4 3 4 2 -4|| EDOs: [[5edo|5]], [[12edo|12]] [[Badness]]: 0.0248 =Injera= [[Comma]]s: 50/49, 81/80 The wedgie for injera is <<2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera. [[POTE tuning|POTE generator]]: 694.375 Map: [<2 0 -8 -7|, <0 1 4 4|] [[Wedgie]]: <<2 8 8 8 7 -4|| EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[140edo|140]], [[178edo|178]] [[Badness]]: 0.0311 =Godzilla= [[Comma]]s: 49/48, 81/80 Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo]] is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes. [[POTE tuning|POTE generator]]: 252.635 Map: [<1 0 -4 2|, <0 2 8 1|] [[Wedgie]]: <<2 8 1 8 -4 -20|| EDOs: [[5edo|5]], [[9edo|9]], [[14edo|14]], [[19edo|19]] [[Badness]]: 0.0267 ==Music== [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3|Godzilla Example]] by [[Cameron Bobro]] [[http://tinyurl.com/4uyumk9|"Change is on the Wind"]] in Godzilla[9] by [[Igliashon Jones]] =Mohajira= [[Comma]]s: 81/80, 6144/6125 Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs. Mohajira can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4. [[7-limit|7]] and [[9-limit]] minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 348.415 Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly. Map: [<1 1 0 6|, <0 2 8 -11|] [[Generator]]s: 2, 128/105 [[Wedgie]]: <<2 8 -11 8 -23 -48|| EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]] [[Badness]]: 0.0557 ==11-limit== [[Comma]]s: 81/80, 121/120, 176/175 [[11-limit]] minimax 1/4 comma [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |6 0 -11/8 0 0>, |2 0 5/8 0 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 348.477 Map: [<1 1 0 6 2|, <0 2 8 -11 5|] [[Generator]]s: 2, 11/9 EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]] [[Badness]]: 0.0261 =Maqamic= [[Comma]]s: 81/80, 36/35, 121/120 Maqamic temperament is a linear mimicry of maqam music within the regular mapping paradigm, much as pelogic temperament is a linear mimicry of the gamelan pelog scale. It is the temperament that you get if you take, at face value, the simplest harmonic mapping consistent with the scalar structure of the maqamat. It deliberately ignores the issue of whether or not those dyads are resolved by the auditory system as such when played melodically (as maqam music is generally a melodic art form), intending instead to serve as a vehicle for the adventurous microtonalist who wants to explore a harmonic context for maqam music. Melodically, it is a "linearized" version of the maqam modal system; it elevates the notion of "the diatonic scale with quartertones" out of the realm of 24-equal into a higher-dimensional rank 2 temperament. If the maqamat are taken to be rank 2, they all end up being MODMOS's of the proper 3L4s MOS. This MOS takes a neutral third as a generator, and the usual 5L2s diatonic scale is itself a MODMOS. The "quarter-tone" chroma in this setup is exactly half of the usual chromatic semitone from 5L2s (regardless of how it's intonated). Harmonically, intervals in this temperament are mapped to reflect some of the intonational choices that actual arabic maqam musicians make when playing maqam music on a fretless instrument such as an oud. Sometimes notes that share the same notational representation are intoned very differently under different circumstances; for example, while the fourths are generally intoned very close to 4/3, it's common to adaptively intone the minor 7th closer to 7/4. When this happens, the two notes are said to share a tempered equivalence class, and the comma between the two versions of the note is said to vanish, no matter how large it is. Real-life intonational differences existing within these equivalence classes can then be viewed as a form of adaptive intonation within this temperament, paralleling how western string quartets will often intone their major chords to be as close to 4:5:6 as possible, despite that 81/80 vanishes over the larger structure of the music. This temperament will hence appear less accurate than it really is, due to the fact that it approximates the performance of highly adaptive fretless instruments, and equivalence classes should be understood as reflecting a cognitive grouping of intervals rather than demanding any particular "middle of the road" intonational ideal. For fixed pitch instruments, 17-equal and 24-equal support this temperament about as well as can be done, but it should be noted that this temperament was designed particularly with adaptive intonation in mind. Like mohajira, maqamic tempers out 121/120 and 81/80; unlike it, it eliminates 36/35 (and hence 64/63) instead of 176/175. Harmonically, maqamic temperament maps two whole tones to 5/4, which is an intonational choice sometimes made by actual Arabic maqam performers, and which indicates that 81/80 vanishes. It also maps two fourths to 7/4, which is likewise an intonational choice often made by maqam performers, although they tend to do this adaptively rather than optimizing their 4/3's around this ideal. Finally, we find the simplest harmonic intervals approximating the neutral second and neutral third, which are 11/10 and 11/9. These, when doubled, yield 6/5 and 3/2, respectively, thus indicating that 121/120 and 243/242 vanish in this tuning. Other attempts to mimic the structure of maqam music within the regular mapping paradigm may exist; this temperament is meant to be the simplest harmonic template possible that is consistent with the scalar structure of maqam music. For example, if one wanted to consider the two neutral thirds and neutral seconds as being different sizes, that would correspond to some sort of rank-3 temperament, and if one wanted to consider the 7/4 as being a "quarter tone" flat of 9/5 (where the quarter tone is the chroma for the 7-note MOS), that would correspond to Mohajira above. [[POTE tuning|POTE generator]]: 350.934 Map: [<1 1 0 4 2|, <0 2 8 -4 5|] [[Generator]]s: 2, 11/9 EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]], [[31edo|31d]] ==13-limit== The 13-limit version of this temperament eliminates 144/143 and hence 169/168 as well; this signifies that the generator could also be taken as (or intoned as) 16/13, and also that the 6/5's, which are also 7/6's, are evenly divided into two equal 13/12's. [[Comma]]s: 81/80, 36/35, 121/120, 144/143 [[POTE tuning|POTE generator]]: 350.816 Map: [<1 1 0 4 2 4|, <0 2 8 -4 5 -1|] Generators: 2, 11/9 EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]],[[31edo| 31d]] =Mothra= [[Comma]]s: 81/80, 1029/1024 Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. [[7-limit|7]] and [[9-limit]] minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 232.193 Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents. Map: [<1 1 0 3|, <0 3 12 -1|] [[Generator]]s: 2, 8/7 [[Wedgie]]: <<3 12 -1 12 -10 -36|| EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]] [[Badness]]: 0.0371 ==11-limit== [[Comma]]s: 81/80, 99/98, 385/384 POTE generator: ~63/55 = 232.031 Map: [<1 1 0 3 5|, <0 3 12 -1 -8|] EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]], [[88edo|88]], [[150edo|150]], [[181edo|181]] [[Badness]]: 0.0256 =Squares= [[Comma]]s: 81/80, 2401/2400 Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third ([[9_7|9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401. 7 and 9 limit minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 425.942 Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents. Map: [<1 3 8 6|, <0 -4 -16 -9|] [[Generator]]s: 2, 9/7 EDOs: [[14edo|14]], [[31edo|31]], [[262edo|262]], [[293edo|293]] [[Badness]]: 0.0460 Music: By [[Chris Vaisvil]] [[http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3|Square 8]] ==11-limit== [[Comma]]s: 81/80, 385/384, 1375/1372 [[POTE tuning|POTE generator]]: 425.993 Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|] EDOs: [[14edo|14]], [[31edo|31]], [[200edo|200]] [[Badness]]: 0.0568 =Liese= [[Comma]]s: 81/80, 686/675 Liese, with wedgie <<3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55. 7 and 9 limit minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |2/3 0 11/12 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 632.406 Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges. Map: [<1 0 -4 -3|, <0 3 12 11|] [[Generator]]s: 2, 10/7 EDOs: [[17edo|17]], [[19edo|19]], [[55edo|55]], [[74edo|74]] [[Badness]]: 0.0467 =Squares= Commas: 81/80, 2401/2400 POTE generator: ~9/7 = 425.942 [[Map]]: [<1 3 8 6|, <0 -4 -16 -9|] [[Wedgie]]: <<4 16 9 16 3 -24|| EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]] [[Badness]]: 0.0460 ==11-limit== Commas: 81/80, 99/98, 121/120 POTE generator: ~9/7 = 425.957 Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|] EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]] [[Badness]]: 0.0216 ==13-limit== Commas: 81/80, 99/98, 121/120, 66/65 POTE generator: ~9/7 = 425.550 Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|] EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]] [[Badness]]: 0.0255 =Jerome= Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5^(1/20), or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size. Commas: 81/80, 17280/16807 POTE generator: ~54/49 = 139.343 Map: [<1 1 0 2|, <0 5 20 7|] Wedgie: <<5 30 7 20 -3 -40|| EDOs: 8, 9, 17, 26, 43, 112 Badness: 0.1087 ==11-limit== Commas: 81/80, 99/98, 864/847 POTE generator: ~12/11 = 139.428 Map: [<1 1 0 2 3|, <0 5 20 7 4|] EDOs: 8, 9, 17, 26, 43, 241 Badness: 0.0479 ==13-limit== Commas: 77/78, 81/80, 99/98, 144/143 POTE generator: ~13/12 = 139.387 Map: [<1 1 0 2 3 3|, <0 5 20 7 4 6|] EDOs: 8, 9, 17, 26, 43, 155, 198 Badness: 0.0293 ==17-limit== Commas: 78/77, 81/80, 99/98, 144/143, 189/187 POTE generator: ~13/12 = 139.362 Map: [<1 1 0 2 3 3 2|, <0 5 20 7 4 6 18|] EDOs: 8, 9, 17, 26, 43, 155 Badness: 0.0209
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<html><head><title>Meantone family</title></head><body><!-- ws:start:WikiTextTocRule:62:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><div style="margin-left: 2em;"><a href="#x-Seven limit children">Seven limit children</a></div> <!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><div style="margin-left: 1em;"><a href="#Septimal meantone">Septimal meantone</a></div> <!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens">Unidecimal meantone aka Huygens</a></div> <!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens-Tridecimal meantone">Tridecimal meantone</a></div> <!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens-Grosstone">Grosstone</a></div> <!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Meanpop">Meanpop</a></div> <!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanpop-13-limit Meanpop">13-limit Meanpop</a></div> <!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Meanenneadecal">Meanenneadecal</a></div> <!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanenneadecal-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 1em;"><a href="#Flattone">Flattone</a></div> <!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 1em;"><a href="#Dominant">Dominant</a></div> <!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 1em;"><a href="#Sharptone">Sharptone</a></div> <!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 1em;"><a href="#Injera">Injera</a></div> <!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 1em;"><a href="#Godzilla">Godzilla</a></div> <!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 2em;"><a href="#Godzilla-Music">Music</a></div> <!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 1em;"><a href="#Mohajira">Mohajira</a></div> <!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 2em;"><a href="#Mohajira-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 1em;"><a href="#Maqamic">Maqamic</a></div> <!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 2em;"><a href="#Maqamic-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 1em;"><a href="#Mothra">Mothra</a></div> <!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 2em;"><a href="#Mothra-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 1em;"><a href="#Squares">Squares</a></div> <!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 2em;"><a href="#Squares-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 1em;"><a href="#Liese">Liese</a></div> <!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 1em;"><a href="#Squares">Squares</a></div> <!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><div style="margin-left: 2em;"><a href="#Squares-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 2em;"><a href="#Squares-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><div style="margin-left: 1em;"><a href="#Jerome">Jerome</a></div> <!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 2em;"><a href="#Jerome-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 2em;"><a href="#Jerome-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 2em;"><a href="#Jerome-17-limit">17-limit</a></div> <!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --></div> <!-- ws:end:WikiTextTocRule:94 -->The <a class="wiki_link" href="/5-limit">5-limit</a> parent <a class="wiki_link" href="/Comma">comma</a> of the <a class="wiki_link" href="/meantone">meantone</a> family is the Didymus or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow">syntonic comma</a>, 81/80. This is the one they all temper out. The <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.239<br /> <br /> <a class="wiki_link" href="/Map">Map</a>: [<1 0 -4|, <0 1 4|]<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/81edo">81</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.00736<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> The <a class="wiki_link" href="/7-limit">7-limit</a> children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1>, |-13 10 0 -1>], flattone, with normal list [|-4 4 -1>, |-17 9 0 1>], dominant, with normal list [|-4 4 -1>, |6 -2 0 -1>], sharptone, with normal list [|-4 4 -1>, |2 -3 0 1>], injera, with normal list [|-4 4 -1>, |-7 8 0 -2>], mohajira, with normal list [|-4 4 -1>, |-23 11 0 2>], godzilla, with normal list [|-4 4 -1>, |-4 -1 0 2>], mothra, with normal list [|-4 4 -1>, |-10 1 0 3>], squares, with normal list [|-4 4 -1>, |-3 9 0 -4>], and liese, with normal list [|-4 4 -1>, |-9 11 0 -3>].<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Septimal meantone"></a><!-- ws:end:WikiTextHeadingRule:2 -->Septimal meantone</h1> The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the <a class="wiki_link" href="/7_4">7/4</a> of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and <a class="wiki_link" href="/7_5">7/5</a>, C-F#, the tritone. The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and <a class="wiki_link" href="/31edo">31edo</a> is a good tuning for it.<br /> <br /> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125<br /> <br /> 7 and <a class="wiki_link" href="/9-limit">9-limit</a> minimax<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.495<br /> <br /> Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.<br /> <br /> <a class="wiki_link" href="/Map">Map</a>: [<1 0 -4 -13|, <0 1 4 10|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<1 4 10 4 13 12||<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/81edo">81</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0137<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Septimal meantone-Unidecimal meantone aka Huygens"></a><!-- ws:end:WikiTextHeadingRule:4 -->Unidecimal meantone aka Huygens</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 99/98<br /> <br /> <a class="wiki_link" href="/11-limit">11-limit</a> minimax<br /> [|1 0 0 0 0>, |25/16 -1/8 0 0 1/16>, |9/4 -1/2 0 0 1/4>,<br /> |21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 11/9<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.967<br /> <br /> <a class="wiki_link" href="/Algebraic%20generator">Algebraic generator</a>: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.<br /> <br /> <a class="wiki_link" href="/Map">Map</a>: [<1 0 -4 -13 -25|, <0 1 4 10 18|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/105edo">105</a>, <a class="wiki_link" href="/198edo">198</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0170<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Tridecimal meantone"></a><!-- ws:end:WikiTextHeadingRule:6 -->Tridecimal meantone</h3> <a class="wiki_link" href="/Comma">Comma</a>s: 66/65, 81/80, 99/98, 105/104<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.642<br /> <br /> Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/267edo">267</a>, <a class="wiki_link" href="/298edo">298</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0180<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Grosstone"></a><!-- ws:end:WikiTextHeadingRule:8 -->Grosstone</h3> Commas: 81/80, 99/98, 126/125, 144/143<br /> <br /> POTE generator: ~3/2 = 697.264<br /> <br /> Map: [<1 0 -4 -13 -25 29|, <0 1 4 10 18 -16|]<br /> EDOs: 12, 31, 43, 74<br /> Badness: 0.0259<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Septimal meantone-Meanpop"></a><!-- ws:end:WikiTextHeadingRule:10 -->Meanpop</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 385/384<br /> <br /> <a class="wiki_link" href="/11-limit">11-limit</a> <a class="wiki_link" href="/minimax">minimax</a> 1/4 comma<br /> [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>,<br /> |-3 0 5/2 0 0>, |11 0 -13/4 0 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.434<br /> <br /> <a class="wiki_link" href="/Algebraic%20generator">Algebraic generator</a>: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.<br /> <br /> Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/81edo">81</a>, <a class="wiki_link" href="/112edo">112</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0215<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="Septimal meantone-Meanpop-13-limit Meanpop"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit Meanpop</h3> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 105/104, 144/143, 196/195<br /> <br /> POTE generator: ~3/2 = 696.211<br /> <br /> Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]<br /> EDOS: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/81edo">81</a>, <a class="wiki_link" href="/131edo">131</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0209<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Septimal meantone-Meanenneadecal"></a><!-- ws:end:WikiTextHeadingRule:14 -->Meanenneadecal</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 81/80<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.250<br /> <br /> Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|]<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/81edo">81</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0214<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="Septimal meantone-Meanenneadecal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit</h3> <a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 78/77, 81/80<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.146<br /> <br /> Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|]<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/131edo">131</a>, <a class="wiki_link" href="/181edo">181</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0212<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h1> --><h1 id="toc9"><a name="Flattone"></a><!-- ws:end:WikiTextHeadingRule:18 -->Flattone</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 525/512<br /> <br /> The <a class="wiki_link" href="/wedgie">wedgie</a> for flattone is <<1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that <a class="wiki_link" href="/7_4">7/4</a> is a diminished minor seventh interval. Other intervals are <a class="wiki_link" href="/7_6">7/6</a>, a diminished minor third, and <a class="wiki_link" href="/7_5">7/5</a>, a doubly diminshed fifth. Good tunings for flattone are <a class="wiki_link" href="/26edo">26edo</a>, <a class="wiki_link" href="/45edo">45edo</a> and <a class="wiki_link" href="/64edo">64edo</a>.<br /> <br /> <a class="wiki_link" href="/7-limit">7-limit</a> minimax<br /> [|1 0 0 0>, |21/13 0 1/13 -1/13>,<br /> |32/13 0 4/13 -4/13>, |32/13 0 -9/13 9/13>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 7/5<br /> <br /> <a class="wiki_link" href="/9-limit">9-limit</a> minimax<br /> [|1 0 0 0>, |17/11 2/11 0 -1/11>,<br /> |24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 693.779<br /> <br /> Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.<br /> <br /> Map: [<1 0 -4 17|, <0 1 4 -9|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<1 4 -9 4 -17 -32||<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/64edo">64</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0386<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h1> --><h1 id="toc10"><a name="Dominant"></a><!-- ws:end:WikiTextHeadingRule:20 -->Dominant</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 36/35, 64/63<br /> <br /> The wedgie for dominant is <<1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is <a class="wiki_link" href="/12edo">12edo</a>, but it also works well with the Pythagorean tuning of pure <a class="wiki_link" href="/3_2">3/2</a> fifths, and with <a class="wiki_link" href="/29edo">29edo</a>, <a class="wiki_link" href="/41edo">41edo</a>, or <a class="wiki_link" href="/53edo">53edo</a>.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.573<br /> <br /> Map: [<1 0 -4 6|, <0 1 4 -2|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<1 4 -2 4 -6 -16||<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/65edo">65</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0207<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h1> --><h1 id="toc11"><a name="Sharptone"></a><!-- ws:end:WikiTextHeadingRule:22 -->Sharptone</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 21/20, 28/27<br /> <br /> Sharptone, with a wedgie <<1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. <a class="wiki_link" href="/12edo">12edo</a> tuning does sharptone about as well as such a thing can be done.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 700.140<br /> <br /> Map: [<1 0 -4 -2|, <0 1 4 3|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<1 4 3 4 2 -4||<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/12edo">12</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0248<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h1> --><h1 id="toc12"><a name="Injera"></a><!-- ws:end:WikiTextHeadingRule:24 -->Injera</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 50/49, 81/80<br /> <br /> The wedgie for injera is <<2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. <a class="wiki_link" href="/38edo">38edo</a>, which is two parallel <a class="wiki_link" href="/19edo">19edo</a>s, is an excellent tuning for injera.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 694.375<br /> <br /> Map: [<2 0 -8 -7|, <0 1 4 4|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<2 8 8 8 7 -4||<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/38edo">38</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/178edo">178</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0311<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h1> --><h1 id="toc13"><a name="Godzilla"></a><!-- ws:end:WikiTextHeadingRule:26 -->Godzilla</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 49/48, 81/80<br /> <br /> Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. <a class="wiki_link" href="/19edo">19edo</a> is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 252.635<br /> <br /> Map: [<1 0 -4 2|, <0 2 8 1|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<2 8 1 8 -4 -20||<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/9edo">9</a>, <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/19edo">19</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0267<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h2> --><h2 id="toc14"><a name="Godzilla-Music"></a><!-- ws:end:WikiTextHeadingRule:28 -->Music</h2> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3" rel="nofollow">Godzilla Example</a> by <a class="wiki_link" href="/Cameron%20Bobro">Cameron Bobro</a><br /> <a class="wiki_link_ext" href="http://tinyurl.com/4uyumk9" rel="nofollow">"Change is on the Wind"</a> in Godzilla[9] by <a class="wiki_link" href="/Igliashon%20Jones">Igliashon Jones</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h1> --><h1 id="toc15"><a name="Mohajira"></a><!-- ws:end:WikiTextHeadingRule:30 -->Mohajira</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 6144/6125<br /> <br /> Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. <a class="wiki_link" href="/31edo">31edo</a> makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.<br /> <br /> Mohajira can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4.<br /> <br /> <a class="wiki_link" href="/7-limit">7</a> and <a class="wiki_link" href="/9-limit">9-limit</a> minimax 1/4 comma<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.415<br /> <br /> Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.<br /> <br /> Map: [<1 1 0 6|, <0 2 8 -11|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 128/105<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<2 8 -11 8 -23 -48||<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/24edo">24</a>, <a class="wiki_link" href="/31edo">31</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0557<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h2> --><h2 id="toc16"><a name="Mohajira-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 121/120, 176/175<br /> <br /> <a class="wiki_link" href="/11-limit">11-limit</a> minimax 1/4 comma<br /> [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>,<br /> |6 0 -11/8 0 0>, |2 0 5/8 0 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.477<br /> <br /> Map: [<1 1 0 6 2|, <0 2 8 -11 5|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 11/9<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/24edo">24</a>, <a class="wiki_link" href="/31edo">31</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0261<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h1> --><h1 id="toc17"><a name="Maqamic"></a><!-- ws:end:WikiTextHeadingRule:34 -->Maqamic</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 36/35, 121/120<br /> <br /> Maqamic temperament is a linear mimicry of maqam music within the regular mapping paradigm, much as pelogic temperament is a linear mimicry of the gamelan pelog scale. It is the temperament that you get if you take, at face value, the simplest harmonic mapping consistent with the scalar structure of the maqamat. It deliberately ignores the issue of whether or not those dyads are resolved by the auditory system as such when played melodically (as maqam music is generally a melodic art form), intending instead to serve as a vehicle for the adventurous microtonalist who wants to explore a harmonic context for maqam music.<br /> <br /> Melodically, it is a "linearized" version of the maqam modal system; it elevates the notion of "the diatonic scale with quartertones" out of the realm of 24-equal into a higher-dimensional rank 2 temperament. If the maqamat are taken to be rank 2, they all end up being MODMOS's of the proper 3L4s MOS. This MOS takes a neutral third as a generator, and the usual 5L2s diatonic scale is itself a MODMOS. The "quarter-tone" chroma in this setup is exactly half of the usual chromatic semitone from 5L2s (regardless of how it's intonated).<br /> <br /> Harmonically, intervals in this temperament are mapped to reflect some of the intonational choices that actual arabic maqam musicians make when playing maqam music on a fretless instrument such as an oud. Sometimes notes that share the same notational representation are intoned very differently under different circumstances; for example, while the fourths are generally intoned very close to 4/3, it's common to adaptively intone the minor 7th closer to 7/4. When this happens, the two notes are said to share a tempered equivalence class, and the comma between the two versions of the note is said to vanish, no matter how large it is. Real-life intonational differences existing within these equivalence classes can then be viewed as a form of adaptive intonation within this temperament, paralleling how western string quartets will often intone their major chords to be as close to 4:5:6 as possible, despite that 81/80 vanishes over the larger structure of the music.<br /> <br /> This temperament will hence appear less accurate than it really is, due to the fact that it approximates the performance of highly adaptive fretless instruments, and equivalence classes should be understood as reflecting a cognitive grouping of intervals rather than demanding any particular "middle of the road" intonational ideal. For fixed pitch instruments, 17-equal and 24-equal support this temperament about as well as can be done, but it should be noted that this temperament was designed particularly with adaptive intonation in mind.<br /> <br /> Like mohajira, maqamic tempers out 121/120 and 81/80; unlike it, it eliminates 36/35 (and hence 64/63) instead of 176/175. Harmonically, maqamic temperament maps two whole tones to 5/4, which is an intonational choice sometimes made by actual Arabic maqam performers, and which indicates that 81/80 vanishes. It also maps two fourths to 7/4, which is likewise an intonational choice often made by maqam performers, although they tend to do this adaptively rather than optimizing their 4/3's around this ideal. Finally, we find the simplest harmonic intervals approximating the neutral second and neutral third, which are 11/10 and 11/9. These, when doubled, yield 6/5 and 3/2, respectively, thus indicating that 121/120 and 243/242 vanish in this tuning.<br /> <br /> Other attempts to mimic the structure of maqam music within the regular mapping paradigm may exist; this temperament is meant to be the simplest harmonic template possible that is consistent with the scalar structure of maqam music. For example, if one wanted to consider the two neutral thirds and neutral seconds as being different sizes, that would correspond to some sort of rank-3 temperament, and if one wanted to consider the 7/4 as being a "quarter tone" flat of 9/5 (where the quarter tone is the chroma for the 7-note MOS), that would correspond to Mohajira above.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 350.934<br /> <br /> Map: [<1 1 0 4 2|, <0 2 8 -4 5|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 11/9<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/10edo">10c</a>, <a class="wiki_link" href="/17edo">17c</a>, <a class="wiki_link" href="/24edo">24d</a>, <a class="wiki_link" href="/31edo">31d</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h2> --><h2 id="toc18"><a name="Maqamic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->13-limit</h2> The 13-limit version of this temperament eliminates 144/143 and hence 169/168 as well; this signifies that the generator could also be taken as (or intoned as) 16/13, and also that the 6/5's, which are also 7/6's, are evenly divided into two equal 13/12's.<br /> <br /> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 36/35, 121/120, 144/143<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 350.816<br /> <br /> Map: [<1 1 0 4 2 4|, <0 2 8 -4 5 -1|]<br /> Generators: 2, 11/9<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/10edo">10c</a>, <a class="wiki_link" href="/17edo">17c</a>, <a class="wiki_link" href="/24edo">24d</a>,<a class="wiki_link" href="/31edo"> 31d</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:38:<h1> --><h1 id="toc19"><a name="Mothra"></a><!-- ws:end:WikiTextHeadingRule:38 -->Mothra</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 1029/1024<br /> <br /> Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using <a class="wiki_link" href="/31edo">31edo</a> with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra.<br /> <br /> <a class="wiki_link" href="/7-limit">7</a> and <a class="wiki_link" href="/9-limit">9-limit</a> minimax 1/4 comma<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 232.193<br /> <br /> Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.<br /> <br /> Map: [<1 1 0 3|, <0 3 12 -1|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 8/7<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<3 12 -1 12 -10 -36||<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/31edo">31</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0371<br /> <br /> <!-- ws:start:WikiTextHeadingRule:40:<h2> --><h2 id="toc20"><a name="Mothra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 99/98, 385/384<br /> <br /> POTE generator: ~63/55 = 232.031<br /> <br /> Map: [<1 1 0 3 5|, <0 3 12 -1 -8|]<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/88edo">88</a>, <a class="wiki_link" href="/150edo">150</a>, <a class="wiki_link" href="/181edo">181</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0256<br /> <br /> <!-- ws:start:WikiTextHeadingRule:42:<h1> --><h1 id="toc21"><a name="Squares"></a><!-- ws:end:WikiTextHeadingRule:42 -->Squares</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 2401/2400<br /> <br /> Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (<a class="wiki_link" href="/9_7">9/7</a>) intervals, and uses it for a generator. <a class="wiki_link" href="/31edo">31edo</a>, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.<br /> <br /> 7 and 9 limit minimax 1/4 comma<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.942<br /> <br /> Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.<br /> <br /> Map: [<1 3 8 6|, <0 -4 -16 -9|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 9/7<br /> EDOs: <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/262edo">262</a>, <a class="wiki_link" href="/293edo">293</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0460<br /> <br /> Music:<br /> By <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3" rel="nofollow">Square 8</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:44:<h2> --><h2 id="toc22"><a name="Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 385/384, 1375/1372<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.993<br /> <br /> Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|]<br /> EDOs: <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/200edo">200</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0568<br /> <br /> <!-- ws:start:WikiTextHeadingRule:46:<h1> --><h1 id="toc23"><a name="Liese"></a><!-- ws:end:WikiTextHeadingRule:46 -->Liese</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 686/675<br /> <br /> Liese, with wedgie <<3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. <a class="wiki_link" href="/74edo">74edo</a> makes for a good liese tuning, though <a class="wiki_link" href="/19edo">19edo</a> can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.<br /> <br /> 7 and 9 limit minimax 1/4 comma<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |2/3 0 11/12 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 632.406<br /> <br /> Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.<br /> <br /> Map: [<1 0 -4 -3|, <0 3 12 11|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 10/7<br /> EDOs: <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/55edo">55</a>, <a class="wiki_link" href="/74edo">74</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0467<br /> <br /> <!-- ws:start:WikiTextHeadingRule:48:<h1> --><h1 id="toc24"><a name="Squares"></a><!-- ws:end:WikiTextHeadingRule:48 -->Squares</h1> Commas: 81/80, 2401/2400<br /> <br /> POTE generator: ~9/7 = 425.942<br /> <br /> <a class="wiki_link" href="/Map">Map</a>: [<1 3 8 6|, <0 -4 -16 -9|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<4 16 9 16 3 -24||<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/8edo">8</a>, <a class="wiki_link" href="/11edo">11</a>, <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/31edo">31</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0460<br /> <br /> <!-- ws:start:WikiTextHeadingRule:50:<h2> --><h2 id="toc25"><a name="Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:50 -->11-limit</h2> Commas: 81/80, 99/98, 121/120<br /> <br /> POTE generator: ~9/7 = 425.957<br /> <br /> Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|]<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/8edo">8</a>, <a class="wiki_link" href="/11edo">11</a>, <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/31edo">31</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0216<br /> <br /> <!-- ws:start:WikiTextHeadingRule:52:<h2> --><h2 id="toc26"><a name="Squares-13-limit"></a><!-- ws:end:WikiTextHeadingRule:52 -->13-limit</h2> Commas: 81/80, 99/98, 121/120, 66/65<br /> <br /> POTE generator: ~9/7 = 425.550<br /> <br /> Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|]<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/8edo">8</a>, <a class="wiki_link" href="/11edo">11</a>, <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/31edo">31</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0255<br /> <br /> <!-- ws:start:WikiTextHeadingRule:54:<h1> --><h1 id="toc27"><a name="Jerome"></a><!-- ws:end:WikiTextHeadingRule:54 -->Jerome</h1> Jerome is related to <a class="wiki_link" href="/20ed5">Hieronymus' tuning</a>; the Hieronymus generator is 5^(1/20), or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.<br /> <br /> Commas: 81/80, 17280/16807<br /> <br /> POTE generator: ~54/49 = 139.343<br /> <br /> Map: [<1 1 0 2|, <0 5 20 7|]<br /> Wedgie: <<5 30 7 20 -3 -40||<br /> EDOs: 8, 9, 17, 26, 43, 112<br /> Badness: 0.1087<br /> <br /> <!-- ws:start:WikiTextHeadingRule:56:<h2> --><h2 id="toc28"><a name="Jerome-11-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->11-limit</h2> Commas: 81/80, 99/98, 864/847<br /> <br /> POTE generator: ~12/11 = 139.428<br /> <br /> Map: [<1 1 0 2 3|, <0 5 20 7 4|]<br /> EDOs: 8, 9, 17, 26, 43, 241<br /> Badness: 0.0479<br /> <br /> <!-- ws:start:WikiTextHeadingRule:58:<h2> --><h2 id="toc29"><a name="Jerome-13-limit"></a><!-- ws:end:WikiTextHeadingRule:58 -->13-limit</h2> Commas: 77/78, 81/80, 99/98, 144/143<br /> <br /> POTE generator: ~13/12 = 139.387<br /> <br /> Map: [<1 1 0 2 3 3|, <0 5 20 7 4 6|]<br /> EDOs: 8, 9, 17, 26, 43, 155, 198<br /> Badness: 0.0293<br /> <br /> <!-- ws:start:WikiTextHeadingRule:60:<h2> --><h2 id="toc30"><a name="Jerome-17-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->17-limit</h2> Commas: 78/77, 81/80, 99/98, 144/143, 189/187<br /> <br /> POTE generator: ~13/12 = 139.362<br /> <br /> Map: [<1 1 0 2 3 3 2|, <0 5 20 7 4 6 18|]<br /> EDOs: 8, 9, 17, 26, 43, 155<br /> Badness: 0.0209</body></html>